2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/105117Let $\De u+\la u=\De v+\la v=0$, where $\De$ is the Laplace--Beltrami operator on a compact connected smooth manifold $M$ and $\la>0$. If $H^1(M)=0$ then there exists $p\in M$ such that $u(p)=v(p)=0$. For homogeneous $M$, $H^1(M)\neq0$ implies the existence of a pair $u,v$ as above that has no common zero.8 pages. This is the published paper with several additional comments in footnotesMetric GeometryAnalysis of PDEs58J50text